On the second boundary value problem for a class of fully nonlinear flows III

We study the solvability of the second boundary value problem of the Lagrangian mean curvature equation arising from special Lagrangian geometry. By the parabolic method we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle-Warren’s theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.